Editing Extendible cardinal (section)

==Properties==
For a cardinal <math>\kappa</math>, say that a logic <math>L</math> is <math>\kappa</math>-compact if for every set <math>A</math> of <math>L</math>-sentences, if every subset of <math>A</math> or cardinality <math><\kappa</math> has a model, then <math>A</math> has a model. (The usual [[compactness theorem]] shows <math>\aleph_0</math>-compactness of first-order logic.) Let <math>L_\kappa^2</math> be the [[infinitary logic]] for second-order set theory, permitting infinitary conjunctions and disjunctions of length <math><\kappa</math>. <math>\kappa</math> is extendible iff <math>L_\kappa^2</math> is <math>\kappa</math>-compact.<ref>{{cite journal
| last1=Magidor | first1=M. | authorlink1=Menachem Magidor
| title=On the Role of Supercompact and Extendible Cardinals in Logic
| date=1971
| pages=147–157
| journal=[[Israel Journal of Mathematics]]
| volume=10
| issue=2
| doi=10.1007/BF02771565 | doi-access=free}}</ref>